Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results
Abstract
:1. Introduction
2. Materials and Methods
2.1. CNNs
2.2. Hopfield NNs
2.3. CNNs with Delays
2.4. CNNs with Reaction–Diffusion Terms
2.5. Cohen–Grossberg DCNNs
2.6. DCNNs with Reaction–Diffusion Terms of Cohen–Grossberg Type
2.7. Bidirectional Associative Memory (BAM) Neural Networks
2.8. Impulsive DCNNs
2.9. Fractional-Order Impulsive CNNs
2.10. Extended Stability Concepts
2.10.1. Practical stability
2.10.2. Stability of Sets
2.10.3. Stability with Respect to Manifolds
2.10.4. Practical Stability with Respect to Manifolds
2.10.5. Lipschitz stability
2.10.6. Lyapunov Approach
3. Results
3.1. Stability of Sets
3.2. Stability with Respect to Manifolds
3.3. Practical Stability with Respect to Manifolds
3.4. Lipschitz Stability
- There exists a continuous for function , , such that
- For ,
4. Discussion
- PS = Practical stability;
- SS = Stability of sets;
- SRhM = Stability with respect to h-manifolds;
- SRIM = Stability with respect to integral manifolds;
- PSRhM = Practical stability with respect to h-manifolds;
- PSRIM = Practical stability with respect to integral manifolds;
- LS = Lipschitz stability;
- DCNNs = Delayed cellular neural networks;
- RDDCNNs = Reaction–diffusion delayed cellular neural networks;
- CGDCNNs = Cohen–Grossberg delayed cellular neural networks;
- RDCGDCNNs = Reaction–diffusion Cohen–Grossberg delayed cellular neural networks;
- BAMDCNNs - BAM delayed cellular neural networks;
- FDCNNs = Fractional delayed cellular neural networks;
- FRDDCNNs = Fractional reaction–diffusion delayed cellular neural networks;
- FCGDCNNs = Fractional Cohen–Grossberg delayed cellular neural networks;
- FRDCGDCNNs = Fractional reaction–diffusion Cohen–Grossberg delayed cellular neural networks;
- FBAMDCNNs = Fractional BAM delayed cellular neural networks.
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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NNs | PS | SS | SRhM | SRIM | PSRhM | PSRIM | LS |
---|---|---|---|---|---|---|---|
DCNNs | √ | √ | √ | √ | √ | √ | √ |
RDDCNNs | √ | √ | √ | √ | √ | × | √ |
CGDCNNs | × | √ | × | √ | × | × | √ |
RDCGDCNNs | × | √ | × | √ | × | × | √ |
BAMDCNNs | √ | × | √ | × | √ | √ | × |
FDCNNs | √ | × | √ | √ | √ | × | √ |
FRDDCNNs | × | × | √ | √ | √ | × | √ |
FCGDCNNs | × | × | × | × | × | × | √ |
FRDCGDCNNs | × | × | × | × | × | × | √ |
FBAMRDDCNNs | × | × | √ | × | × | × | × |
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Stamov, G.; Stamova, I. Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results. Fractal Fract. 2023, 7, 289. https://doi.org/10.3390/fractalfract7040289
Stamov G, Stamova I. Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results. Fractal and Fractional. 2023; 7(4):289. https://doi.org/10.3390/fractalfract7040289
Chicago/Turabian StyleStamov, Gani, and Ivanka Stamova. 2023. "Extended Stability and Control Strategies for Impulsive and Fractional Neural Networks: A Review of the Recent Results" Fractal and Fractional 7, no. 4: 289. https://doi.org/10.3390/fractalfract7040289